**INFORMATION ABOUT PROJECT,
SUPPORTED BY RUSSIAN SCIENCE FOUNDATION**

*The information is prepared on the basis of data from the information-analytical system RSF, informative part is represented in the author's edition. All rights belong to the authors, the use or reprinting of materials is permitted only with the prior consent of the authors. *

COMMON PART

Project Number21-12-00409

Project titleCritical phenomena in mesoscopic many-body systems and their applications to state-of-art computer technologies

Project LeadTarasov Sergei

AffiliationThe Institute for Physics of Microstructures - branch of Federal research center A.V. Gaponov-Grekhov Institute of Applied Physics of the Russian Academy of Sciences,

Implementation period | 2021 - 2023 |

Research area 02 - PHYSICS AND SPACE SCIENCES, 02-201 - Theory of condensed matter

KeywordsSuperconductivity, spin systems, disorder, Bose-Einstein condensate, ultracold atoms, Ising model, subgap states, superconducting heterostructures, violation of Eigenstate thermalization hypothesis, quantum memory on disordered systems; nonergodic extended phases, multifractality Anderson and many-body localization

PROJECT CONTENT

Annotation

Critical phenomena are the attributes of state-of-art and future technologies. Already now magnetic-structure based memory is being used in every computer.
Further miniaturization of transistors leads to the overheating problem of high-performance CPUs due to the Joule heating in semiconducting and metallic parts. Thus, more and more the demand on dissipationless technologies based on superconductors grows.
Disorder and impurities also play an important role in the manufacturing process of computing elements. On one hand, detrimental effects of impurities, which leads to the scattering of charge carriers, transport suppression and carrier mobility reduction, affect the efficiency and characteristics of such devices.
However, on the other hand, in combination with superconducting or magnetic order parameters, disorder opens new possibilities, e.g., for the increase of the superconducting critical currents, allows the superconductors to work in moderate magnetic fields with help of the pinning of Abrikosov vortices. Moreover the impurities themselves may work as traps for hot quasiparticles and, thus, increase decoherence time of superconducting devices for quantum computing.
In addition, strong disorder effects contribute to the creation of the localized phases in spin systems, where non-equilibrium states may be realized and have very long lifetime. Such structures are good for quantum memory applications.
Thus, the investigation of critical phenomena in superconducting, spin, and disordered quantum systems to which the current project is devoted to has straightforward applications to state-of-art quantum and classical computing.
Additional investigation of many-body disordered systems as well as random-matrix models and hierarchical structures as quantum simulators of main aspects of many-body localization (MBL) in the systems with interactions (superconducting pairing, spin exchange, or Coulomb interparticle interaction) is needed for the description of spectral, transport, and dynamical characteristics of such systems
Taking into account effects of isolated defects on superconducting properties of homogeneous superconductors or Josephson junctions opens possibilities for controllable superconductivity with the tunable energy spectrum.
Although the project is purely theoretical, most planned problems can be directly verified in experiments of leading international groups.
Advancement in the above mentioned research directions requires using the entire arsenal of methods of theoretical physics. We will apply both standard methods of microscopic superconductivity theory (including non-stationary and/or quasiclassical parts), renormalization group, multifractal analysis of wave functions and numerical calculations of disordered systems via exact diagonalization and shift-inverse iteration. Moreover, in some cases it will be useful to apply some methods and measures from quantum information theory (e.g., entanglement entropy growth with subsystem size or evolution time) and recently developed generalized Anderson and Mott’s principles for determination of ergodic and localization transitions in the systems with strong correlations. The key to success of the project is the balanced composition of personnel, including both active young researchers (including Bachelor and PhD students) and experienced scientists who can teach young participants and prepare them for the solution of ambitious problems of the project.

Expected results

One can distinguish three main research directions of the project:
I) Effects of isolated impurities and fluctuations of the order parameter on the properties of condensed mesoscopic systems in applications to classical and quantum computing.
More and more complicated tasks that humanity posts in front of the computers force year-by-year decrease the size of transistors in classical processors. On one hand, this leads to the overheating and heat discharge problems in such systems. On the other hand, it raises the question of quantum coherent and mesoscopic effects (related to finite and small system sizes). The latter is related to the mesoscopic fluctuations, especially close to the phase transitions. Moreover the classical computer architecture itself brings constraints to their computing capabilities, e.g., on the exponential times needed for solving NP-complete problems.
For both resolving the overheating problem and quantum computer realizations more and more scientists and engineers pay great attention to the dissipationless transport and coherence preserving in superconductors and superconducting hybrid structures. Here the main role starts to be played by the control of the transport of superfluid carries. In our project, we suggest using isolated impurities for such a control.
Thus, within the project
- Effects of isolated impurities on subgap superconducting states with a multiquantum vortex will be investigated within the applications to realization of the hot quasiparticle traps. The calculation will be done for finite temperatures.
- Effects of an isolated impurity on the subgap spectrum of the superconductor/insulator/superconductor Josephson junction will be investigated, including the case of the presence of a Josephson vortex.
Solving these problems will allow us to clarify how isolated impurities affect transport and spectral properties of superconducting structures and to work out recommendations for utilization of defects for elements of superconducting electronics.
Another aspect of miniaturization of modern computer elements is the increasing role of the mesoscopic fluctuations in such systems. Such effects are especially strong and important close to the phase transition (whether systems permit them). In order to understand fundamental limitations on usage of the mesoscopic systems for the realization of the classical computing elements as well as for the understanding of perspectives of their utilization as quantum simulators and calculators we plan to investigate the microscopic theory of critical phenomena, which is applicable to the phase transitions of the same universality class (namely to the Bose-Einstein condensation of interacting atoms and superconductivity, as well as to the magnetic transitions in mesoscopic lattice spin models). This theory will allow describing the vicinity of the phase transition critical point, where the order parameter fluctuations are essential and the constraints that are put on the many-body Hilbert space by the particle conservation in the system are important.
Such methods related to the fundamental physical problem of the phase transition description appear to be on demand as well for realizations of computing devices based on the small-size systems with phase transition (such as magnetic memory cells). The point is that now the description of the properties of the systems in the vicinity of a critical point is a complicated task (as the mean-field theory breaks down in this region). For moderate systems typical for the current applications, the critical region is quite narrow, thus the operating states of the device appear to be away from the phase transition. Miniaturization of such devices leads to the broadening of the critical region to the wider parameter range, thus the possible overlap of it with the device operation range might be inevitable. Moreover quantum computations based on mesoscopic systems in the critical range of parameters are of particular interest and might appear to be in the most demand due to their natural relation to the non-trivial #P- and NP-complete properties of investigated quantum mesoscopic systems.
In addition, the combination of the above mentioned superconducting and magnetic order parameters with the ideas of the individual control of the system properties by the isolated impurities opens new horizons for the applications and universal quantum computations. In particular, the system of magnetic adatoms on a superconductor is known to be a platform for the realization of the topologically protected Majorana modes. However an important aspect of any electronics is the tunable character of its elements. Thus within the project
- We plan to calculate the spectrum of impurity states induced by a chain of magnetic atoms on top of a Josephson junction. The possibility of switching the impurity spectrum between topological and trivial states via the topological quantum phase transition with help of the change of the phase difference will be considered. The phenomenon of the gap closing in the vicinity of the transition will be analyzed.
II) Effects of non-Gaussian fluctuations and emerging fractality in applications to quantum annealing, machine learning, and realization of analog quantum simulators.
The breakdown of the central limit theorem and the existence of non-Gaussian fluctuations often go hand-in-hand with the interaction or hierarchical structure of the Hilbert space in quantum systems. Non-Gaussianity might appear due to both the application of the confinement potential (like in the case of ultracold atoms and ions in traps) and forming of long-ranged (multiqubit) hops in spin systems.
On the other hand, the presence of the hierarchical structure of the Hilbert space or disorder effects might lead to the self-similarity and fractality effects.
Besides the fundamental interest both non-Gaussian fluctuation and the emergent fractality have practical applications: namely, in modern quantum computing for both quantum annealing and machine learning the algorithms for efficient population transfer of the wave packet from a single to the extensive number of local rarely and remotely distributed minima of a potential profile are on high demand. Furthermore not only classical, but also quantum algorithms have exponential complexity for such problems. Recently for a parametric speed-up of such algorithms, it has been suggested to use particularly prepared systems with robust truly multifractal eigenstates. Thus, the suggestion of easily realizable models with robust multifractality are on high demand in this direction.
Basing on the high demand in the quantum computing and machine learning communities within the project
- the known Rosenzweig-Porter random matrix model where the project PI with collaborators has shown the existence of a whole fractal phase will be generalized beyond the self-averaging to the model with non-Gaussian (lognormal) distribution of off-diagonal elements;
- It will be shown that the structure of the robust non-ergodic extended phase in such a model possesses a true multifractality both in the coordinate and energy spaces due to non-Gaussian fluctuations;
- the multifractal properties of eigenstates will be investigated both in the coordinate and energy spaces (showing the structure of multifractal energy minibands).
- the correspondence between the above model and the Anderson model on the hierarchical structures, used for the description of the many-body localization (MBL) transition. The latter connects this project direction to the next one.
- As another realization of the ergodicity breaking a cold (partly condensed) Bose-gas with the fixed number of interacting particles will be considered; the statistics of population of condensate and the excitations for different types of traps; in particular, the joined probability distributions of excitation populations will be investigated;
- It is assumed that the solution of such a fundamental problem for interacting systems will allow us to find out the auto-model (weak multifractality) property in the bulk of the above probability distributions, which has been discovered by some of the project members with coauthors in ideal Bose-gases
- In such Bose systems the optimization of parameters essential for the validity or the breakdown of the central limit theorem in the thermodynamic limit (such as the interaction amplitude and trap potential profile) will be performed. For some particular cases of the systems with non-Gaussian statistics of condensate population the corresponding probability distributions will be calculated.
- As the next step, for the description of the deviations from the ergodicity of the states in the middle of the spectrum for a wide class of the non-integrable disordered systems in the delocalized phase the influence of the mobility edge will be considered;
- The general principle based on the orthogonality of the mid-spectrum states to the special edge spectral ones will be derived and verified on the examples of hierarchical structures (like a random regular graph relevant for the MBL description) and known long-ranged random matrix models with the Anderson transition (like lognormal Rosenzweig-Porter and a paradigmatic power-law random banded matrix (PLRBM) models).
- The analysis of the weak ergodicity breaking and anomaly slow (subdiffusive) transport in such systems will be performed;
- The correspondence of the properties of such systems to the ones of the many-body non-integrable systems for the quantitative description of their properties (like MBL transition) will be investigated.
For more deep understanding of the nature of the above anomaly slow (subdiffusive) transport in disordered systems close to the localization transition and of the relation between subdiffusion and weak ergodicity breaking
- we plan to extract the Thouless time from the spectral properties (like a spectral form factor and the level statistics) of a random regular graph, believed to be a proxy for the Hilbert space of interacting systems undergoing MBL transition;
- the above Thouless time will be compared to the one taken from the wave packet propagation in the same system.
The above single-particle models, in addition, can be considered as quantum simulators (or analog quantum evaluators). Random matrix models may simulate the dynamics of many-body disordered systems which will be used in the last project direction.
- Based on the mesoscopic system of Bose atoms in a trap or spins in a lattice the boson sampling schemes or other analog quantum calculations will be suggested. These schemes lean on the fact that microscopic description of such complicated systems inevitably involves NP- or #P-complex mathematical objects.
- the search of concrete NP- or #P-complex problems and algorithms and protocols which give the solutions of such problems with help of the universal quantum computer based on the reference to the so-called quantum oracle and providing the result of the evolution of one of the above mesoscopic systems, will be performed.
Solution of the problems of this direction shows the possible deviations from ergodicity in both clean and disordered many-body systems (in their delocalized phase).
In addition, the realization of robust nonergodic phases of matter has straightforward applications to the speed-up of machine learning algorithms.
The problems of this direction are closely related to the next one (in particular, in the strong disorder limit) and will be considered synthetically in both research directions from the complemented points of view.
III) Many-body localization and the breakdown of Eigenstate thermalization hypothesis in applications for the realization of quantum memory.
Many-body localization (MBL) prevents quantum thermalization in a generic isolated quantum system (with a strong disorder) and keeps the information about their initial state for infinitely long times. Besides the fundamental interest in the nature of the MBL transition, this phenomenon has immediate relation to the realization of long-lived quantum memory.
- First, within the project we plan to develop both numerically and phenomenologically the relation of the many-body eigenstate properties at the MBL transition in the coordinate basis to wave function structure in the Hilbert space on the example of a paradigmatic quantum Ising model proven to have an MBL transition.
- The relation between the wave function structure in the Hilbert space and the emergent local integral of motions will be investigated;
- The phenomenological model based on the avalanche of ergodic blocks which describes the above relation will be developed.
As the rigorous analytical description of MBL transition is hardy accessible, an important direction for the understanding of the nature of the MBL is the creation of different quantum simulators for this many-body problem. For the description of the extended eigenstates with Poisson level statistics relevant for the MBL phase,
- we plan to develop the effects of partial or full correlations in long-ranged hopping terms in disordered random matrix models on their spectral and localization properties;
- we suppose these models to show Poisson level statistics for the fractal extended eigenstates in the Hilbert space relevant for MBL and usually absent in single-particle models.
- a full analytical and numerical description is planned to be given for the models.
- For more deep understanding of the origin of level repulsion (deviation from Poisson statistics), in particular, in open quantum systems we plan to consider the statistics of exceptional points of two Hermitian Hamiltonian models H1 and H2 both of orthogonal or unitary symmetry coupled via complex term z as follows H1+z H2. The relation between the above statistics and level repulsion is planned to be uncovered for random matrix and local many-body systems.
- As the next step in characterization of the properties of MBL transition we plan to suggest a local disordered interacting fermionic model obeying duality with respect to the Fourier transform and investigate its localization properties. The presence of the self-dual point assures the MBL transition, while the concurrence between the diagonal disorder, hopping and interaction terms presumably provide a non-integrable ETH regime on the one side of the transition.
- For more deep understanding of the mechanism of avalanche destroying MBL state in disordered interacting systems we plan to consider in detail the melting of ergodic blocks and forming the avalanche.
- The main focus will be given to possible non-ergodic character of forming avalanche in the Hilbert space and its relation to slow dynamics of the system.
Solution of the above problems will allow us to more deeply understand the nature of the MBL phase and the transition to this phase.
In addition, it will help us to understand the relation between the eigenstate structures in the coordinate and Hilbert space bases of many-body disordered systems and opens the way for the realization of quantum memory based on such systems.
The suggested problem is quite ambitious and is related to important and actual questions of critical phenomena. Amount of planned work, to our mind, is adequate to the balanced personnel structure which combines young active researchers with two representatives of experienced scientists and premises the success and feasibility of the project. Arguments for this are the existing backlog of the personnel and experience in such research directions, publication activity of a group in the last 3 years, involvement of Bachelor and PhD students in the project and active collaboration with leading Russian and international research groups in the topic.
Based on the results of the mentioned works, a series of articles will be published (approximately 12 publications for 3 years) in the journals Physical Review B (impact factor 3.767), Physical Review Letters (impact factor 7.943), JETP Letters (impact factor 1.52), JETP (impact factor 0.543) and other peer reviewed journals, confirming their compliance with the world standards of research in the given field.

REPORTS

Annotation of the results obtained in 2023

1. To interpret possible experiments on the observation of Majorana states in magnetic chains, the spectrum of subgap states localized on a chain of magnetic atoms with a spiral magnetic order located on the surface of a three-dimensional superconductor was theoretically studied. It has been shown that in the general case the spectrum of an infinite chain contains two bands. When the spins of magnetic atoms lie in a plane, the spectrum is symmetrical with respect to the zero energy level and almost certainly has a gap. Such spin structure is most favorable for the observation of Majorana states. It has been shown that no more than 4/λ subgap states per unit length of the magnetic chain are formed, where λ is the Fermi wavelength. Conditions for the existence of Majorana states at the ends of the chain have been obtained analytically.
2. The asymptotic behavior of wave functions of Majorana states in semi-infinite Kitaev chains, in which the hopping and pairing amplitudes are real and decrease proportionally to the distance between nodes to the powers of α and β, respectively, was studied within the framework of discrete Bogolyubov-de Gennes equations. Using the Wiener-Hopf technique, it has been shown that for α>1 and β>1 the wave function of the Majorana mode decreases with distance l from the edge of the chain proportionally to l to the power -δ, where δ=min(α,β). For α<1 and α<β, the wave function decreases in proportion to l to the power (α-3)/2 – a larger value of the parameter α corresponds to a lower decay rate of the wave function. For such parameters, the energy of the edge mode in a finite chain is inversely proportional to the length of the chain.
3. Within the framework of the Bogolyubov-de Gennes equations, two problems on the influence of mesoscopic effects on the Yu-Shiba-Rusinov states induced by a magnetic atom have been solved. Sub-gap states in a one-dimensional superconducting wire containing two point defects – a magnetic and a non-magnetic one – have been studied. It has been shown that in the limiting case, when the Fermi wavelength and the distance between defects are much smaller than the coherence length, two subgap states are formed. The presence of a non-magnetic defect can lead to a shift of the energies of Yu-Shiba-Rusinov states induced by a magnetic defect by a value of the order of the superconducting gap. We also studied the Yu-Shiba-Rusinov states in a two-dimensional superconducting disk with a point magnetic defect located in the center of the disk. It has been found that in a disk with a radius much smaller than the coherence length there are no subgap states. In general, there are ranges of disk radii in which there are two subgap states, and which alternate with ranges where there are no subgap states. Thus, the edge of the superconductor significantly affects the Yu-Shiba-Rusinov states, as do non-magnetic defects.
4. We have finished the study of the vortex pinning mechanisms in type-II superconductors with modulated disorder. During the reporting period we have investigated the influence of the predicted switching in the vortex pinning mechanism on transport properties of such structures. For rather strong magnetic fields the superconducting state of such systems is rather a set of superconducting droplets coupled by the Josephson interaction. Within the Ginzburg-Landau-type theory with modulated diffusion coefficient we have derived analytic expressions for the Josephson energy and the critical supercurrent for a pair of identical overlapping superconducting nuclei with unit and zero vorticity. The increase in the external magnetic field leads to the switching between the above-mentioned states and, thus, can lead to the change in the slope of the critical current.
5. Analysis of localization and fractality has been performed in a so-called beta-ensemble of tridiagonal matrices with independent random matrix elements, distributed in such a way that the level repulsion of such matrices is the same as in Gaussian random matrix ensembles, but with any (not only integer) parameter beta. Such analysis has also been performed in the models with correlations in diagonal disorder potential. It has been shown that in the beta-ensemble there coexist localized and extended states in the same energy interval of the spectrum without any mobility-edge formation. Main criteria to realize this scenario have been developed. For small values of beta <<1 the above-mentioned tridiagonal matrix model has been mapped locally (in a vicinity of each site) to the one-dimensional Anderson model with the hopping amplitude, scaling as a square root of the site index.
6. Formation of a mobility edge has been investigated in the Anderson model on the random regular graph with only finite fraction of nodes, subject to the disorder. The robustness of the above mobility edge has been analyzed with respect to any amplitude of the diagonal disorder. An effective model for Green’s function calculations on the disorder-free nodes has been derived. This model has been shown to become asymptotically exact at large disorder amplitudes. The critical fraction of disordered nodes has been calculated, below which the delocalized states survive.
7. The computational complexity has been investigated for matrix permanents and hafnians which describe – according to the methods we developed previously – statistical characteristics of such many-particle quantum systems as a thermally equilibrium cold interacting Bose gas and a lattice of spins exhibiting Ising-type interactions. We have found configurations of these systems in which the effective, polynomial-time calculation of their statistical characteristics by classical computers appears significantly difficult. Such configurations arise when the contribution of anomalous correlators to the values of involved hafnians are relatively large. For the atomic bosonic sampling process, in which hafnians of different matrices define joint probabilities of the outcomes of simultaneous measurement of the numbers of atoms in a given set of spatial modes, a parameter has been identified that largely determines the complexity of the classical simulation.
8. The generalization of the graphical method for the calculation of the spectrum of fractal dimension of the local density of states has been performed. This method, developed in 2022 by the investigators of the current project, for describing the generalized Rosenzweig-Porter (RP) models, has been now generalized to the case of correlated real and imaginary parts of Green’s function self-energies in the cavity method. The absence of multifractality has been analytically proven in a generic case and shown numerically for so-called Levy-RP и LN-RP models. The only sparse Erdos-Renyi-RP models cannot be described by the above method. The generalization of this method to the finite-size systems has been analyzed and the methods to reduce finite-size effects have been developed.
9. The generic method of phase-diagram calculation has been developed for the random long-range models with the correlated diagonal disorder, when the correlations are translation-invariant in the site space. The phase diagram, level repulsion as well as fractal dimensions of eigenstates have been calculated across the disorder amplitudes. The above-mentioned analytical method has been applied to the set of exemplary systems with power-law decaying hopping amplitudes and power-law correlated diagonal disorder. Such a set of models has been shown to be self-dual with respect to the spatial Fourier transform. The applicability of the method, described above, has been confirmed.
10. The eigenstate properties on random regular graphs with the node number N, vertex degree d and the enhanced number of short loops of the length k have been calculated. The number of k-loops has been controlled via the chemical potential μ_k>0, deforming the distribution of such graphs over the entire set of random regular graphs. The localization phase diagram has been numerically and analytically analyzed in the parameter plane (μ_k,d) for different N. The critical curves, separating 4 main phases, have been calculated. The above phases are: homogeneous (unclustered), ideally and non-ideally clusterized, as well as the scarred unclustered one. It has been shown that in the latter phase it is topologically equivalent nodes (TENs) that play the main role for the formation of scars, similar to those in the interacting many-body systems. At the same time, these TENs are nuclei of the clusterization process.
11. The localization phase diagram of a Floquet-driven one-dimensional Aubry-Andre model has been calculated. The result has been shown to strongly depend on the order of limits: thermodynamic limit of infinite system sizes N and the drive period T. The increase of T has been shown to increase the effective localization length in the model (in the real or momentum space depending on the drive parameters). This localization length has been shown to be drive-period dependent and growing with it, but staying finite (i.e. cannot make the system really diffusive) at any finite T. It has been shown that for the drive parameters, mixing the localized phases in real and momentum space at different parts of the drive period, the thermodynamic limit of infinite N leads to the formation of multiple mobility edges in the spectrum.

Publications

**1.** *Kochergin D., Khaymovich I., Valba O., Gorsky A.* **Robust extended states in Anderson model on partially disordered random regular graphs** SciPost Physics, SciPost Phys. 16, 106 (2024) (year - 2024) https://doi.org/10.21468/SciPostPhys.16.4.106

**2.** *Kutlin A.G., Khaymovich I.M.* **Anatomy of the eigenstates distribution: A quest for a genuine multifractality** SciPost Physics, том 16, номер 008 (year - 2024) https://doi.org/10.21468/SciPostPhys.16.1.008

**3.** *Bespalov A.* **Majorana edge states in Kitaev chains of the BDI symmetry class** SciPost Physics Core, Т. 6, № статьи 80 (year - 2023) https://doi.org/10.21468/SciPostPhysCore.6.4.080

**4.** *D. Kochergin, I. M. Khaymovich, O. Valba, A. Gorsky* **Anatomy of the fragmented Hilbert space: eigenvalue tunneling, quantum scars and localization in the perturbed random regular graph** Physical Review B, том 108, номер 09, страница 094203 (year - 2023) https://doi.org/10.1103/PhysRevB.108.094203

**5.** *Das. A. K.,Ghosh A., Khaymovich I. M.* **Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States** Physical Review Letters, том 131, выпуск 16, страница 166401 (year - 2023) https://doi.org/10.1103/PhysRevLett.131.166401

**6.** *Gonçalves M.,Ribeiro P., Khaymovich I. M.* **Quasiperiodicity hinders ergodic Floquet eigenstates** Physical Review B, том 108, выпуск 10, страница 104201 (year - 2023) https://doi.org/10.1103/PhysRevB.108.104201

**7.** *Kopasov A.A., Tsar'kov I.M., Mel'nikov A.S.* **Disorder-induced trapping and antitrapping of vortices in type-II superconductors** Physical Review B, том 107, номер 17, страница 174505 (year - 2023) https://doi.org/10.1103/PhysRevB.107.174505

**8.** *Bespalov A.A.* **Волновые функции майорановских состояний в цепочках Китаева симметрийного класса BDI** Нанофизика и наноэлектроника. Труды XXVII Международного симпозиума (Нижний Новгород, 13–16 марта 2023 г.), Т. 1, с. 16-17 (year - 2023)

**9.** *Khaymovich I. M., De Tomasi G.* **Усиление локализации в неэрмитовых дальнодействующих неупорядоченных моделях** Нанофизика и наноэлектроника. Труды XXVII Международного симпозиума (Нижний Новгород, 13–16 марта 2023 г.), том 1, страницы 119-120 (year - 2023)

**10.** *Mishin A.V., Kocharovsky Vl.V., Kocharovskaya E.R.* **Поляритонные моды в запрещенной фотонной зоне сверхизлучающего гетеролазера. Сравнительный анализ при наличии и в отсутствие инверсии населенностей** Нанофизика и наноэлектроника. Труды XXVII Международного симпозиума (Нижний Новгород, 13–16 марта 2023 г.), Т. 2, с. 687-688 (year - 2023)

**11.** *Tarasov S.V., Kocharvosky Vl.V.* **Совместная статистика чисел частиц в наборе возбужденных мод бозе-конденсата и ее вычислительная сложность** Нанофизика и наноэлектроника. Труды XXVII Международного симпозиума (Нижний Новгород, 13–16 марта 2023 г.), Т. 1, с. 111-112 (year - 2023)

Annotation of the results obtained in 2021

1. The energy spectrum of Bogolyubov quasiparticles localized on a chain of magnetic impurities with a spiral magnetic order, placed inside a three-dimensional Josephson junction, has been calculated. The band spectrum of quasiparticles has been obtained analytically for an infinite chain. The parameters of the system have been determined for which its state is topologically nontrivial. It was demonstrated that the topological state of the system can be controlled by changing the Josephson phase difference. For the topologically nontrivial state, the wave functions of Majorana fermions, localized at the ends of a finite chain, have been calculated.
2. We develop a method of describing Bose condensation in a mesoscopic system containing a finite and fixed number of interacting atoms, which implies an initial consideration of a simpler auxiliary problem with unlimited occupation numbers and the subsequent calculation of the characteristics of the original problem in terms of the Green's functions of the auxiliary system. Within the framework of the developed description, it is shown that the characteristic function describing the probability distributions of the occupation numbers of the excited modes is expressed in terms of the correlation matrix composed of the expected values of the pairs of creation and annihilation operators of the complete set of excited single-particle states. We have also shown that if one measures simultaneously the occupation numbers of various excited modes of the considered Bose system, the probabilities of outcomes are connected to the NP-hard complexity class characteristics of special matrices, related to the correlation matrix mentioned above.
3. For an equilibrium low-temperature Bose gas of interacting particles confined in a mesoscopic trap with an arbitrary external potential profile, we calculate explicitly the characteristic function which determines the probability distribution of the order parameter, i.e. the total number of particles in the condensate. Sufficient conditions for the inapplicability of the central limit theorem to the description of the order parameter fluctuations are formulated. Namely, we have shown that in the thermally dominated regime, the statistical distribution for the number of condensed particles in a weakly interacting gas are non-Gaussian, if the corresponding ideal-gas problem of finding of the ground state occupation statistics, whose solution is known, leads to non-Gaussian fluctuations in a case of the same trap with a potential modified by the developed condensate.
4. Both statically and dynamically the Rosenzweig-Porter model with logarithmically-normal distribution of off-diagonal elements (as we called LN-RP further) has been investigated.
The existence of not only fractal (like in the conventional Gaussian Rosenzweig-Porter model), but genuinely multifractal static phase has been shown to exist versus the disorder strength and the strength of fluctuations (parameterized by the relative width p of the log-normal distribution).
Within the Wigner-Weisskopf approximation, both mean and typical return probability of a wave packet initialized on a single site has been investigated. The existence of exponential, stretch-exponential and more slow (polynomial) decay of the mean return probability, corresponding to diffusive, sub-diffusive and frozen-dynamics phases, has analytically been found.
The case of the random regular graph, corresponding to p=1 on the LN-RP phase diagram, has been shown to demonstrate a tricritical point, where simultaneously a static multifractal phase disappears, while a dynamical frozen-dynamics phase emerges. This analysis leads to the fact that at any finite size both the random regular graph and LN-RP model show multifractal behavior close to the Anderson transition (and it has been numerically confirmed). However, this multifractality is just a finite-size effect in this case. This investigation resolves long-standing debates on this issue present in the literature since 2011.
Results of this work have been published as I. M. Khaymovich, V. E. Kravtsov, “Dynamical phases in a "multifractal" Rosenzweig-Porter model”, SciPost Phys. 11(2), 045 (2021) (https://scipost.org/SciPostPhys.11.2.045) [arXiv:2106.01965].
5. The structure of a wave function in coordinate and Hilbert space has been investigated across the many-body localization (MBL) transition on the example of a disordered quantum Ising model. A new sensitive physical quantity for the description of the MBL transition, namely, a probability distribution \Pi(x) of Hamming distance x in a many-body wave function with respect to its maximum, has been suggested. The relations of fractal dimensions D_q of a wave function in the Hilbert space and the localization length \xi of real-space local integrals of motion, emerging in the MBL phase, to the above-mentioned distribution \Pi(x) have been uncovered. Jumps in D_q and \xi at the MBL transition have been shown to be determined by the presence of a corresponding jump in the mean <x> over the distribution \Pi(x). The jumps in all three physical quantities are shown to originate from the breakdown of self-averaging at the MBL transition, unlike both thermalizing and MBL phase. The matching of the above observations has been found to be consistent with a phenomenological scenario of the “avalanche”, which is accepted in the literature for the description of the MBL phase transition.
Results of this work have been published as G. De Tomasi, I. M. Khaymovich, F. Pollmann, S. Warzel,
“Rare thermal bubbles at the many-body localization transition from the Fock space point of view”,
Phys. Rev. B 104, 024202 (2021) (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.024202),
Editor’s Suggestion [arXiv:2011.03048].
6. The effects of partial correlations in kinetic hopping terms of long-range disordered random matrix models on their localization properties have been studied. A set of models considered in the problem interpolates from fully - correlated Richardson’s model to the random Rosenzweig-Porter model without correlations. Both analytically and numerically it has been shown that any deviation from the completely correlated case leads to the emergent non-ergodic delocalization in the system unlike the predictions of the localization by the cooperative shielding method. In order to describe the models with correlated kinetic terms, we develop the generalization of the Dyson Brownian motion and the cavity method approaches and examine its applicability to the above set of models.
Results of this work have been published as A. G. Kutlin, I. M. Khaymovich, “Emergent fractal phase in energy stratified random models,” SciPost Physics 11, 6, 101 (2021).
7. Within the phenomenological Ginzburg-Landau theory we have studied the distinctive features of the superconducting phase transition in the systems with modulated disorder in the presence of an applied magnetic field. We predict the appearance of the disorder induced first-order phase transition between different vortex phases and the tricritical point at the phase diagram magnetic field - temperature.
8. Within the supersymmetric approach, the statistics of Green’s functions between distant sites in a hierarchical structure of the Cayley tree has been calculated, where the Hamming distance is of the order of the diameter. The moments of Green’s function have been shown to be determined by a maximal eigenvalue of a linearized transfer-matrix problem on the same graph. The results of the forward scattering approximation, considered to be exact at the infinitely strong disorder, have been shown to not be applicable even at strong disorder to the Anderson model on the Cayley tree. The cases of finite-dimensional lattices have been discussed.
Results of calculations have been put on a repository (arXiv:2108.10326) and are under consideration in the journal SciPost Physics (https://scipost.org/submissions/2108.10326v2/).
9. The possibility to realize delocalized states with Poisson energy-level statistics has been investigated. The main principles to find a system with the phase, where the bulk eigenstates are non-ergodic and delocalized, while the corresponding eigenvalues do not repulse each other, have been developed.

Publications

**1.** *A. G. Kutlin, I. M. Khaymovich* **Emergent fractal phase in energy stratified random models** SciPost Physics, том 11, номер 6, страница 101 (year - 2021) https://doi.org/10.21468/SciPostPhys.11.6.101

**2.** *Giuseppe De Tomasi, Ivan M. Khaymovich, Frank Pollmann, and Simone Warzel* **Rare thermal bubbles at the many-body localization transition from the Fock space point of view** Physical Review B, том 104, номер 02, страница 024202 (year - 2021) https://doi.org/10.1103/PhysRevB.104.024202

**3.** *I. M. Khaymovich, V. E. Kravtsov* **Dynamical phases in a "multifractal" Rosenzweig-Porter model** SciPost Physics, том 11, номер 2, страница 045 (year - 2021) https://doi.org/10.21468/SciPostPhys.11.2.045

Annotation of the results obtained in 2022

The paper, devoted to the calculations of the spectrum of impurity states, induced by a magnetic-adatom chain, which is placed in a Josephson contact (problem 1 in 2021), has been published:
A. A. Bespalov “Tuning the topological state of a helical atom chain via a Josephson phase”,
Phys. Rev. B 106, 134503 (2022) https://doi.org/10.1103/PhysRevB.106.134503.
1. The investigation of the Majorana states in a Kitaev chain with the real-valued coefficients of a generic spatial profile have been performed.
Such a system belongs to the BDI universality class according to the Altland-Zirnbauer classification of ensembles [C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016)]. Using the Wiener-Hopf technique, an exact analytical solution for the wave functions of zero-energy Majorana eigenstates has been obtained in a semiinfinite chain.
2. The effects of fluctuations of isolated-state energies of Yu-Shiba-Rusinov states in a magnetic atom chain, placed inside a Josephson junction on the quasiparticle states with non-zero energies have been investigated. It is shown that in the presence of the above-mentioned disorder all the subgap states with the energies, small compared to the superconducting gap, become Anderson localized. The spatial profile of such localized states resembles the one of the Majorana modes in a clean case.
3.We have developed a method for the microscopic description of an arbitrary lattice of interacting spins in a thermal equilibrium state. The method is based on the connections between the original spin system, described via the Holstein-Primakov representation in terms of spin bosons with restricted excitation numbers, and an auxiliary problem of interacting bosonic modes with unbounded occupation numbers. The statistical characteristics of the system of spins — the partition function, the average magnetization of a lattice site, and pairwise correlations of the magnetization — have been expressed in terms of the joint probabilities of the distribution, characterizing occupation numbers for the auxiliary bosonic problem. This distribution has been found explicitly via the method of the characteristic function. We have shown that the characteristic function of the obtained functional form corresponds to the joint probabilities proportional to the matrix Hafnians. Calculating Hafnian of a general case matrix belongs to the #P-complexity class, and this fact gives interesting insights on possible applications of spin systems in quantum computing.
An article connecting the characteristic function of the obtained type with explicit analytical expressions for joint probabilities has been accepted for publication in the journal Q1
V. V. Kocharovsky, Vl. V. Kocharovsky, S. V. Tarasov, PRA accepted 17.11.2022: https://journals.aps.org/pra/accepted/3707dYf3Ab615e6a68623b136e2c7b2a71606f435.
4. The effect of switching of a vortex-pinning mechanism in type-II superconductors with modulated disorder has been predicted. It is shown that in weak enough magnetic fields the spatial regions with a reduced diffusion coefficient attract vortices, while the magnetic-field increase may push the vortices away from such defects. Such an effect has been investigated both for the case of isolated defects and for the superconductors with the regular lattices of such defects. On an example of a regular defect lattice, it has been demonstrated that the predicted switching of a vortex-pinning mechanism reveals itself in the transformation of the vortex-lattice structure in strong magnetic fields.
A. A. Kopasov, I. M. Tsar’kov, and A. S. Mel’nikov, Disorder-induced trapping and anti-trapping of vortices in type-II superconductors, arXiv:2211.13531 (https://arxiv.org/abs/2211.13531).
5. The Thouless time of the transport of excitations through the system, given by the Anderson model on a random regular graph, has been calculated from the global spectral characteristics (such as a spectral form-factor and a power spectrum) both in static and periodically driven cases. The calculation of both quantities has confirmed a subdiffusive scaling of the Thouless time with the graph diameter L in both static and driven models. The above results have been also compared to the with the ones, obtained in the corresponding random matrix ensemble of the Rosenzweig-Porter model with the log-normal distribution of the off-diagonal elements (LNRP) and it has been confirmed that the subdiffusive character of the wave-packet dynamics in the Anderson model on the random regular graph survives till very small, but finite disorder amplitudes, in agreement with the predictions, done in the LNRP model.
6. The task about the mapping of the above-mentioned LNRP model to the Anderson model on the hierarchical structures (such as the random regular graph or the Cayley tree) has been finished. Within a supersymmetric field-theoretic approach, the Green's function moments in the Anderson model on the Cayley tree have been calculated between two distant points at the distance of the order of the graph diameter. It has been shown that such moments may be written in terms of the maximal eigenvalue of a linearized problem of the transfer matrix on the same graph. Both the forward scattering approximation (FSA), and the log-normal distributions of the above Green's functions have been shown to give inaccurate approximations, the deviations from the exact solution has been shown to grow with both the graph diameter and the distance r between the points on the graph.
The results of this work have been published in a Q1 journal:
P. A. Nosov, I. M. Khaymovich, A. Kudlis, V. E. Kravtsov
“Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation”, SciPost Phys. 12, 048 (2022) https://scipost.org/SciPostPhys.12.2.048 [arXiv:2108.10326].
7. The main conditions on a system to realize a phase of matter, possessing the majority of eigenstates to be non-ergodic and extended, but no level repulsion of eigenenergies (Poisson level statistics) have been determined.
As an example, a generalization of a so-called Burin-Maksimov model with the off-diagonal matrix elements, scaled up as a power N^{\gamma} of a system size N, has been considered.
A renormalization group approach to solve the problem has been analytically developed and used in a combination with a so-called coherent potential approximation for the global density of states in order to find a spatial structure of mid-spectral eigenstates. This spatial structure has been shown to be given by the exponential decay with a certain localization length \xi_l, which grows as a power-law with the system size, \xi_l ~ N^{\gamma}, followed at larger distances by a standard polynomial localized tail, known from the Burin-Maksimov model at \gamma=0.
The results of this work have been published in the Quantum journal (Q1):
W. C. Tang, I. M. Khaymovich, “Non-ergodic delocalized phase with Poisson level statistics”, Quantum 6, 733 (2022) https://doi.org/10.22331/q-2022-06-09-733, [arXiv:2112.09700].
8. The mechanisms of the emergence of significantly multifractal (but not just fractal) states in random-matrix proxies of many-body systems have been investigated. In particular, the generalized Rosenzweig-Porter (RP) models, with non-Gaussian off-diagonal-element distribution (e.g., Levy-RP or Log-normal RP) has been considered. In order to analytically describe the statistics of eigenstates, the (graphical) method for calculation of the spectral of fractal dimensions of the wave-function amplitudes has been developed, based on a so-called cavity method. As a result of the application of the above method, we have to conclude that there are no phases of multifractal states in the generalized RP models. Another important result is that the significant ingredient for the realization of a multifractality is the correlation between the off-diagonal matrix elements, absent in the RP models.
9. The investigation of the Yu-Shiba-Rusinov states, induced by a magnetic adatom on a surface of a 3D superconductor, has been performed. The magnetic adatom is considered as a P-scatterer, i.e., the scattered wave by this atom has an orbital moment equal to l=1. In addition, the spin-up and spin-down electrons acquire different phases via scattering. It is shown that such a magnetic adatom induces two Yu-Shiba-Rusinov subgap states, similarly to a point-like impurity in a 3D superconductor. Using the Green’s function technique, the eigenenergies and eigenfunctions of these states have been calculated.
10. The general principle of the deviation of the entanglement entropy from its ergodic value in generic local many-body systems, related to the orthogonality of the mid-spectrum states to the special low-entanglement states at the spectral edges, has been developed. It is shown that this orthogonality leads to the entanglement-entropy deviations from the Page value entanglement. Numerical simulations confirm the above analytical prediction: the entanglement-entropy deviations are indeed finite (and survive in the thermodynamic limit), it is parametrically larger than the statistical fluctuations in a finite-size systems and it is directly related to the orthogonality to the spectral-edge states.
The results of this work have been published in Physical Review E (Q1) and available there with the open access:
M. Haque, P. McClarty, I. M. Khaymovich,
“Entanglement of mid-spectrum eigenstates of chaotic many-body systems—deviation from random ensembles.” Phys. Rev. E 105, 014109 (2022) https://doi.org/10.1103/PhysRevE.105.014109.
11. The stability of the localization beyond the convergence of the locator-expansion perturbation theory in quantum systems with long-range (e.g., dipole-dipole) interaction and dilute excitations has been investigated.
It has been shown that such a localization is stable with respect to both the increase of the system dimensionality from d=1 to d=2 and to a small, but finite anisotropy (given by a uniform tilt of all dipoles away from a normal to the 2d plane where they are located).
The analytical explanation of the above phenomena is given in both frameworks of a spatial renormalization group and independently via a so-called matrix-inversion trick, developed in 2019 by one of the main investigators of the current project. In addition, in the framework of Ioffe-Regel localization criterion it has been given an estimate of the measure of the delocalized states and it was confirmed both numerically and analytically that this measure goes to zero in the thermodynamic limit.
The results of this work have been published in the Q1 journal with the open access:
X. Deng, A. L. Burin, I. M. Khaymovich, “Anisotropy-driven localization transition in quantum dipoles”,
SciPost Phys. 13, 116 (2022) https://scipost.org/SciPostPhys.13.5.116.

Publications

**1.** *Bespalov A.A.* **Tuning the topological state of a helical atom chain via a Josephson phase** Physical Review B, т. 106, с. 134503 (year - 2022) https://doi.org/10.1103/PhysRevB.106.134503

**2.** *Deng X., Khaymovich I. M., Burin A. L.* **Anisotropy-mediated reentrant localization** SciPost Physics, том 13, номер 05, страница 116 (year - 2022) https://doi.org/10.21468/SciPostPhys.13.5.116

**3.** *Haque M., McClarty P. A., Khaymovich I. M.* **Entanglement of midspectrum eigenstates of chaotic many-body systems: Reasons for deviation from random ensembles** Physical Review E, том 105, номер 14, страница 014109 (year - 2022) https://doi.org/10.1103/PhysRevE.105.014109

**4.** *Kocharovsky V.V., Kocharovsky Vl.V., Tarasov S.V.* **Atomic boson sampling in a Bose-Einstein condensed gas** Physical Review A, том 106, номер 06, стр. 063312 (year - 2022) https://doi.org/10.1103/PhysRevA.106.063312

**5.** *Nosov P. A. , Khaymovich I. M., Kudlis A., Kravtsov V. E.* **Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation** SciPost Physics, том 12, номер 02, страница 048 (year - 2022) https://doi.org/10.21468/SciPostPhys.12.2.048

**6.** *Tang W., Khaymovich I.M.* **Non-ergodic delocalized phase with Poisson level statistics** Quantum, том 6, номер -, страница 733 (year - 2022) https://doi.org/10.22331/q-2022-06-09-733

**7.** *Bespalov A.A.* **Управление топологическим состоянием цепочки магнитных примесей в сверхпроводнике посредством джозефсоновской разности фаз** Нанофизика и наноэлектроника. Труды XXVI Международного симпозиума (Нижний Новгород, 14–17 марта 2022 г.), Т. 1, с. 42 (year - 2022)

**8.** *Tarasov S.V., Kocharovsky Vl.V., Kocharovsky V.V.* **Гауссова и негауссова статистика флуктуаций бозе-эйнштейновского конденсата в мезоскопической системе** Нанофизика и наноэлектроника. Труды XXVI Международного симпозиума (Нижний Новгород, 14–17 марта 2022 г.), Том 1, стр. 139-141 (year - 2022)

**9.** *Tsar’kov I.M., Kopasov A.A., Mel’nikov A.S.* **Vortex phase transitions in disordered superconductors with spatially modulated diffusion coefficient** Нанофизика и наноэлектроника. Труды XXVI Международного симпозиума (Нижний Новгород, 14–17 марта 2022 г.), Т.1, стр. 26 (year - 2022)